Parametric Differentiation

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Simon Thomas

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Simon Thomas 5 years, 2 months ago

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Simon Thomas 5 years, 2 months ago

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Name	Status	Author	Last Modified
Parametric Differentiation	draft	Simon Thomas	12/02/2020 10:50
Equation of tangent to Parametric curve	draft	Simon Thomas	12/02/2020 10:51
Parametric 2nd derivative	draft	Simon Thomas	12/02/2020 10:52

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Question statement

A curve is defined by the parametric equations $x=\var{a}+\var{b}t, y = \var{c}+\var{d}t^2$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
b	integer	5
c	integer	4
d	integer	5
x	integer	1
t	rational	-1/5

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Generated value: `integer`

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"Unnamed part" → "Gap 3." - Minimum accepted value
"Unnamed part" → "Gap 3." - Maximum accepted value

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1. Gap Gap 0. Number entry
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2. Gap Gap 1. Mathematical expression
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3. Gap Gap 2. Mathematical expression
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4. Gap Gap 3. Number entry
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Ask the student a question, and give any hints about how they should answer this part.

(i) Find $\frac{dx}{dt}$ Gap 0.

(ii) Find $\frac{dy}{dt}$ Gap 1.

(iii) Hence find $\frac{dy}{dx}$ in terms of $t$ Gap 2.

(iv) Hence find $\frac{dy}{dx}$ when $x = \var{x}$ Gap 3.

Advice

We have $x=\var{a}+\var{b}t, y = \var{c}+\var{d}t^2$

(i) To find $\frac{dx}{dt}$ we simply differentiate $x=\var{a}+\var{b}t$ with respect to $t$ to obtain $\frac{dx}{dt}=\var{b}$

(ii) To find $\frac{dy}{dt}$ we simply differentiate $y=\var{c}+\var{d}t^2$ with respect to $t$ to obtain $\frac{dy}{dt}=2\times\var{d}t=\var{2d}t$

(iii) Since $\frac{dy}{dx}=\frac{dy}{dt}/\frac{dx}{dt}$ we have $\frac{dy}{dx}=\frac{\var{2d}t}{\var{b}}=\var{2d/b}t$

(iv) When $x=\var{x}$ , we can use $x=\var{a}+\var{b}t$ to obtain $\var{x}=\var{a}+\var{b}t$ , hence $t = \frac{\var{x}-\var{a}}{\var{b}}=\var{(x-a)/b}$

Substituting this value of $t$ into $\frac{dy}{dx}=\var{2d/b}t$ gives $\frac{dy}{dx}=\var{2d/b}\times \var{(x-a)/b}=\var{2d*(x-a)/b^2}$

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This question is used in the following exam:

Parametric Differentiation by Simon Thomas in Maths support.

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Simon Thomas 5 years, 2 months ago

Simon Thomas 5 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Parametric Differentiation

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 5 years, 2 months ago

Simon Thomas 5 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`